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Decomposition of global solutions for a class of nonlinear wave equations

Georgios Mavrogiannis, Avy Soffer, Xiaoxu Wu

TL;DR

This work studies global solutions to the nonlinear wave equation \Box u = N(x,t,u)u in \mathbb{R}^n$ (n \ge 3) under general space-time dependent nonlinear interactions. The authors construct free channel wave operators and show that, when the interaction is localized, global solutions decompose asymptotically into a free wave plus a localized remainder, with quantitative bounds on the localized part, thereby initiating a multi-channel scattering framework for NLW. The analysis hinges on microlocalization via propagation observables, dispersive estimates for the free flow, and Cook-type representations to control the nonlinear contribution, yielding existence and alpha-independent limits for the channel operators. The results extend earlier scattering theory for Schrödinger equations to a broad class of nonlinear wave equations, providing a rigorous description of asymptotic dynamics and offering a first step toward a full multi-channel scattering theory for NLW with general interactions.

Abstract

In the present paper we consider global solutions of a class of non-linear wave equations of the form \begin{equation*} \Box u= N(x,t,u)u, \end{equation*} where the nonlinearity~$ N(x,t,u)u$ is assumed to satisfy appropriate boundedness assumptions. Under these appropriate assumptions we prove that the free channel wave operator exists. Moreover, if the interaction term~$N(x,t,u)u$ is localised, then we prove that the global solution of the full nonlinear equation can be decomposed into a `free' part and a `localised' part. The present work can be seen as an extension of the scattering results of~\cite{SW20221} for the Schrödinger equation.

Decomposition of global solutions for a class of nonlinear wave equations

TL;DR

This work studies global solutions to the nonlinear wave equation \Box u = N(x,t,u)u in \mathbb{R}^n$ (n \ge 3) under general space-time dependent nonlinear interactions. The authors construct free channel wave operators and show that, when the interaction is localized, global solutions decompose asymptotically into a free wave plus a localized remainder, with quantitative bounds on the localized part, thereby initiating a multi-channel scattering framework for NLW. The analysis hinges on microlocalization via propagation observables, dispersive estimates for the free flow, and Cook-type representations to control the nonlinear contribution, yielding existence and alpha-independent limits for the channel operators. The results extend earlier scattering theory for Schrödinger equations to a broad class of nonlinear wave equations, providing a rigorous description of asymptotic dynamics and offering a first step toward a full multi-channel scattering theory for NLW with general interactions.

Abstract

In the present paper we consider global solutions of a class of non-linear wave equations of the form \begin{equation*} \Box u= N(x,t,u)u, \end{equation*} where the nonlinearity~ is assumed to satisfy appropriate boundedness assumptions. Under these appropriate assumptions we prove that the free channel wave operator exists. Moreover, if the interaction term~ is localised, then we prove that the global solution of the full nonlinear equation can be decomposed into a `free' part and a `localised' part. The present work can be seen as an extension of the scattering results of~\cite{SW20221} for the Schrödinger equation.
Paper Structure (27 sections, 18 theorems, 227 equations)

This paper contains 27 sections, 18 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Background and the aim of this paper
  3. Discussion on our results
  4. Comparison with relevant work
  5. Some useful spaces and operators
  6. The assumptions of the main Theorems
  7. Examples
  8. Preliminary discussion of our Theorem \ref{['main theorem 1']}
  9. Free channel wave operators
  10. The main theorems
  11. Auxiliary technical results I: Lemma \ref{['lem: decay']}, Lemma \ref{['rep: ut']}
  12. The free wave equation
  13. Duhamel formulas
  14. Auxiliary technical results II: (Relative)-Propagation estimates
  15. Preliminary Commutator estimate
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $1\leq q\leq \infty$ be a positive extended real quantity. Let $j$ and $m$ be non-negative integers such that $j<m$. Furthermore, let $1\leq r\leq\infty$ be a positive extended real quantity, $p\geq 1$ be real and $\theta\in [0,1]$ such that the relations hold. Then, for any $u\in L^q(\mathbb R^n)$ such that $D^mu \in L^r(\mathbb R^n)$, with two exceptional cases:

Theorems & Definitions (46)

  • Theorem 1.1: classical Gagliardo-Nirenberg
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • Remark 1.1
  • Definition 1.3
  • ...and 36 more